In Situ Optimization of an Optoelectronic Reservoir Computer with Digital Delayed Feedback

TL;DR

This paper presents an in situ optimization method for an optoelectronic delay-based reservoir computer with digital delayed feedback, enabling real-time tuning of five hyperparameters to optimize performance. By evaluating on waveform classification, NARMA10 time-series prediction, and Japanese vowels classification, the approach yields NMSE values of $0.028$, $0.561$, and $0.271$, respectively, outperforming simulation-based optimization in two tasks. The study also validates that resonances between delay $\tau$ and clock cycle $T$ can degrade accuracy, highlighting the importance of hardware-aware tuning. Overall, the work demonstrates a practical path to deploying high-performance, energy-efficient RC hardware without reliance on extensive software simulations, broadening the applicability of neuromorphic optoelectronic systems.

Abstract

Reservoir computing (RC) is an innovative paradigm in neuromorphic computing that leverages fixed, randomized, internal connections to address the challenge of overfitting. RC has shown remarkable effectiveness in signal processing and pattern recognition tasks, making it well-suited for hardware implementations across various physical substrates, which promise enhanced computation speeds and reduced energy consumption. However, achieving optimal performance in RC systems requires effective parameter optimization. Traditionally, this optimization has relied on software modeling, limiting the practicality of physical computing approaches. Here, we report an \emph{in situ} optimization method for an optoelectronic delay-based RC system with digital delayed feedback. By simultaneously optimizing five parameters, normalized mean squared error (NMSE) of 0.028, 0.561, and 0.271 is achieved in three benchmark tasks: waveform classification, time series prediction, and speech recognition outperforming simulation-based optimization (NMSE 0.054, 0.543, and 0.329, respectively) in the two of the three tasks. This method marks a significant advancement in physical computing, facilitating the optimization of RC and neuromorphic systems without the need for simulation, thus enhancing their practical applicability.

Figures (7)

• Figure 1: a Generic reservoir computing principle. The depicted layout consists of distinct components: an input layer (bronze spheres) responsible for receiving external data, a reservoir (ruby spheres) featuring randomized fixed connections, and a linear readout layer (green spheres). b Delay reservoir computing. $f(x)$ is the activation function performing the nonlinear transformation exhibited by the element, and $h(t)$ is the impulse response. $W$, $W^I$ and $W^R$ are, respectively, reservoir, input, and readout connectivity matrices.
• Figure 2: a. An artistic impression of the experimental set-up. A continuous wave (CW) laser beam is directed into the fiber polarization controller (FPC), which aligns the light polarization with the slow axis of the modulator. Subsequently, the laser beam is coupled to a fiber amplifier to maintain the system's stability. This amplified laser light is modulated with a Mach-Zehnder modulator (MZM), whose sinusoidal transmission function introduces nonlinearity into the reservoir. The modulated light is detected by a photodetector, delayed with Moku:Go's FPGA-based delay, amplified, and sent to the modulation input of the MZM, forming a closed loop. The external signal is mixed with the delayed feedback. Note: the components are out of scale for visualization. b. Photograph of the experimental setup, components are labeled as follows: 1. continuous-wave (CW) laser, 2. Erbium-doped fiber amplifier (EDFA), 3. function generator producing trigger signal. 4. fiber polarization controller (FPC), 5. Mach-Zehnder modulator (MZM) with electrical driver, 6. photodetector, 7. Moku:Go, 8. power supply for the electrical driver, 9. arbitrary waveform generator (AWG), 10. oscilloscope, 11. variable optical attenuator (VOA). c. Schematics of the digital delay and data digitizer implemented in the Moku:Go component: delayed feedback implements the reservoir's connectivity matrix $W$ while the digitized reservoir states are weighted with the readout matrix $W^R$.
• Figure 3: Sinusoidal vs. rectangular waveform classification task. The solid grey curve represents the input signal (sinusoidal and rectangular waveforms), a solid blue curve represents the target response ($x$=1 for sinusoidal, $x$=0 for rectangular waveform), the dash-dotted yellow and orange dashed curves represent the readout of simulated and in situ-optimized experimental reservoirs, respectively. Experimental reservoir settings: input scaling $\rho$=0.19, net gain $G$=0.39, phase bias $\Phi_0$=$0.67\pi$, delay $\tau$=$0.27T$, regularization parameter $\lambda=1.4\times10^{-3}$.
• Figure 4: NARMA10 time series recovery. The solid grey curve represents the input signal (white noise), the solid blue curve represents the time series governed by the NARMA10 model (Eq. (\ref{['eq:narma10']})), the dash-dotted yellow and dashed orange curves represent the readout of simulated and in situ-optimized reservoirs, respectively. Experimental reservoir settings: input scaling $\rho$=0.33, net gain $G=0.7$, phase bias $\Phi_0=0.68\pi$, delay $\tau=0.49T$, regularization parameter $\lambda=5\times10^{-3}$.
• Figure 5: Japanese vowels classification task. Multiplexed waveforms of a. input data, b. ground truth, c-d simulated and experimental reservoirs' readouts. Reservoir settings: input scaling $\rho$=$0.47$, net gain $G$=$0.52$, phase bias $\Phi_0$=$0.44\pi$, delay $\tau$=$0.35T$, regularization parameter $\lambda=3\times10^{-7}$.